Jan 18, 2008 - University of California - Santa-Cruz, Santa Rosa, CA. 14th â 18th ... Miss-steering of photon beams, modification of the dispersion function ...
Linear imperfections and correction Imperfections, USPAS, January 2008
Y. Papaphilippou CERN United States Particle Accelerator School, University of California - Santa-Cruz, Santa Rosa, CA 14th – 18th January 2008
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Linear Imperfections and correction Steering error and closed orbit distortion
Imperfections, USPAS, January 2008
Gradient error and beta beating correction Linear coupling and correction Chromaticity 2
Beam stability in storage rings Beam orbit stability very critical especially for the stability of the synchrotron light spot in the beam lines Consequences of orbit distortion
Miss-steering of photon beams, modification of the dispersion function, resonance excitation, aperture limitations, lifetime reduction, coupling of beam motions, modulation of lattice functions, poor injection efficiency
Long term Causes (Years - months) Imperfections, USPAS, January 2008
Ground settling, season changes, diffusion, medium - Days/Hours, sun and moon, day-night variations (thermal), rivers, rain, water table, wind, synchrotron radiation, refills and start-up, sensor motion, drift of electronics, local machinery, filling patterns
Short (Minutes/Seconds)
Ground vibrations, power supplies, injectors, insertion devices, air conditioning, refrigerators/compressors,water cooling, beam instabilities in general 3
Closed orbit distortion Causes Dipole
field errors Dipole misalignments Quadrupole misalignments Consider the displacement of a particle δx from the ideal orbit . The vertical field is
Imperfections, USPAS, January 2008
quadrupole dipole
Remark: Dispersion creates a closed orbit distortion for off-momentum particles Effect of orbit errors in any multi-pole magnet Feed-down
2(n+1)-pole
2n-pole
2(n-1)-pole
dipole4
Effect of single dipole kick • Introduce Floquet variables
Imperfections, USPAS, January 2008
• • • •
The Hill’s equations are written The solutions are the ones of an harmonic oscillator Consider a single dipole kick at φ=π Then and and in the old coordinates
Maximum distortion amplitude
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Example: Orbit distortion in ESRF storage ring
Imperfections, USPAS, January 2008
In the ESRF storage ring, the beta function is 1.5m in the dipoles and 30m in the quadrupoles. Consider dipole error of δy’=1mrad The horizontal tune is 36.44 Maximum orbit distortion in dipoles
For quadrupole displacement with 1mm, the distortion is Magnet alignment is critical
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Imperfections, USPAS, January 2008
Closed orbit distortion
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Many orbit errors’ effect Consider random distribution of errors in N magnets The expectation value is given by
Example: Imperfections, USPAS, January 2008
In
the ESRF storage ring, there are 64 dipoles The expectation value of the orbit distortion in the dipoles is 4mm, considering a 1mrad dipole error The expection value for the orbit distortion created by the 112 focusing quadrupoles, considering an equivalent 0.1mrad dipole error from misalignments (~ 0.1mm for 1m long quads of 10T/m gradient), is 11mm! 8
Transport of orbit distortion due to dipole kick • Consider a transport matrix between positions 1 and 2
• The transport of transverse coordinates is written as
Imperfections, USPAS, January 2008
• Consider a single dipole kick at position 1 • Then, the first equation may be rewritten • Replacing the coefficient from the general betatron matrix
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Correcting the orbit distortion
Imperfections, USPAS, January 2008
Place horizontal and vertical dipole correctors close to the corresponding quads Simulate (random distribution of errors) or measure orbit in Beam position monitors (downstream of the correctors) Minimize orbit distortion with several methods Globally Harmonic , which minimizes components of the orbit frequency response after a Fourier analysis Most efficient corrector (MICADO), finding the most efficient corrector for minimizing the rms orbit Least square fitting Locally
Sliding Bumps Singular Value Decomposition (SVD) 10
Orbit bumps
Imperfections, USPAS, January 2008
2-bump: Only good for phase advance equal π between correctors Sensitive to lattice and BPM errors Large number of correctors
3-bump: works for any lattice Need large number of correctors No control of angles
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Imperfections, USPAS, January 2008
Singular Value Decomposition example
M. Boege, CAS 2003
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Orbit feedback
Imperfections, USPAS, January 2008
Closed
orbit stabilization performed using slow and fast orbit feedback system. Slow operates every a few seconds (30 for the ESRF) and uses complete set of BPMs (~200 at the ESRF) for both planes Efficient in correcting distortion due to current decay in magnets or other slow processes Fast orbit correction system operates in a wide frequency range (.1Hz to 150Hz for the ESRF) correcting distortions induced by quadrupole and girder vibrations. Local feedback systems used to damp horizontal orbit distortion in ID straight sections, especially the ones were beam spot stabilization is critical
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Gradient error and optics distortion Key issue for the performance -> super-periodicity preservation -> only structural resonances excited Broken super-periodicity -> excitations of all resonances Causes
Imperfections, USPAS, January 2008
Errors
in quadrupole strengths (random and systematic) Injection elements Higher-order multi-pole magnets and errors
Observables Tune-shift Beta-beating Excitation
of integer and half integer resonances 14
Gradient error Consider the transfer matrix for one turn
Imperfections, USPAS, January 2008
Consider a gradient error in a quad. In thin element approximation the quad matrix with and without error are
The new 1-turn matrix is which yields
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Gradient error and tune-shift Consider a new matrix after 1 turn with a new tune
Imperfections, USPAS, January 2008
The traces of the two matrices describing the 1-turn should be equal which gives Developing the left hand side
and finally For a quadrupole of finite length, we have 16
Gradient error and beta distortion Consider the unperturbed transfer matrix for one turn
Introduce a gradient perturbation between the two matrices
Imperfections, USPAS, January 2008
Recall that term as
and write the perturbed
On the other hand and Equating the two terms and integrating through the quad 17
Imperfections, USPAS, January 2008
Example: Gradient error in the ESRF storage ring Consider 128 focusing arc quads in the SNS ring with 0.1% gradient error. In this location =30m. The length of the quads is around 1m The tune-shift is
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Random gradient error distribution For a random distribution of errors the beta
Imperfections, USPAS, January 2008
variation (beating) is
Optics functions beating >10% by putting random errors (0.1% of the gradient) in high dispersion quads of the ESRF storage ring Justifies the choice of quadrupole corrector strength
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Imperfections, USPAS, January 2008
Gradient error correction Windings on the core of the quadrupoles or individual quadrupole correctors (TRIM) Simulation by introducing random distribution of quadrupole errors Compute the tune-shift and the optics function beta distortion Move working point close to integer and half integer resonance Minimize beta wave or quadrupole resonance width with TRIM windings To correct certain resonance harmonics N, strings should be powered accordingly Individual powering of TRIM windings can provide flexibility and beam based alignment of BPM 20
Linear coupling Betatron motion is coupled in the presence of skew quadrupoles
Imperfections, USPAS, January 2008
The field is and Hill’s equations are coupled Motion still linear with two new eigen-mode tunes, which are always split. In the case of a thin quad: Coupling coefficients As motion is coupled, vertical dispersion and optics function distortion appears Causes: Random rolls in quadrupoles Skew quadrupole errors 21 Off-sets in sextupoles
Vertical emittance Ideally the vertical dispersion is zero and the vertical dispersive emittance is zero, so the vertical emittance Real Lattice: Errors as sources of vertical emittance Vertical dipoles (a1) and skew quadrupoles (a2) Dipole and quadrupole rolls Quadrupole and Sextupole vertical displacement
Imperfections, USPAS, January 2008
Result to vertical dispersion and linear coupling needing orbit correctors and skew quadrupoles for suppression Emittance ratio Coupling corrected when coupling coefficient drops to 10−3 level Brightness proportional to the inverse of the coupling coefficient only for hard X-rays (diffraction limitation) Touschek lifetime proportional to
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Imperfections, USPAS, January 2008
Linear coupling correction strategy Introduce skew quadrupole correctors Simulation by introducing random distribution of quadrupole errors Correct globally/locally coupling coefficient (or resonance driving term) Correct optics distortion (especially vertical dispersion) Move working point close to coupling resonances and repeat 23
Example: Coupling correction for the ESRF ring
Imperfections, USPAS, January 2008
Local decoupling using 16 skew quadrupole correctors and coupled response matrix reconstruction Achieved correction of below 0.25% reaching vertical emittance of below 10pm
R. Nagaoka, EPAC 2000 24
Imperfections, USPAS, January 2008
Chromaticity Linear equations of motion depend on the energy (term proportional to dispersion) Chromaticity is defined as: Recall that the gradient is This leads to dependence of tunes and optics function on energy For a linear lattice the tune shift is: So the natural chromaticity is: 25
Example: Chromaticity in the SNS ring
Imperfections, USPAS, January 2008
In the SNS ring, the natural chromaticity is –7. Consider that momentum spread The tune-shift for off-momentum particles is
In order to correct chromaticity introduce particles which can focus off-momentum particle Sextupoles 26
Chromaticity from sextupoles The sextupole field component in the x-plane is: In an area with non-zero dispersion Than the field is quadrupole
dipole
Imperfections, USPAS, January 2008
Sextupoles introduce an equivalent focusing correction The sextupole induced chromaticity is The total chromaticity is the sum of the natural and sextupole induced chromaticity 27
Chromaticity correction
Imperfections, USPAS, January 2008
Introduce sextupoles in high-dispersion areas (not easy to find) Tune them to achieve desired chromaticity Two families are able to control horizontal and vertical chromaticity Sextupoles introduce non-linear fields (chaotic motion) Sextupoles introduce tune-shift with amplitude Example: The ESRF ring has natural chromaticity of -130 Placing 32 sextupoles of length 0.4m in locations where β=30m, and the dispersion D=0.3m For getting 0 chromaticity, their strength should be
or a gradient of 280 T/m2 28
Eddy current sextupole component Sextupole component due to Eddy currents in an elliptic vacuum chamber of a pulsing dipole
with
Imperfections, USPAS, January 2008
Taking into account
with
we get 29
Measurements vs Theory Example: ESRF booster chromaticity Booster Chromaticity without correction 5
Imperfections, USPAS, January 2008
Chromaticity
-5
-15
-25 Horizontal (measurement) Vertical (measurement)
-35
Horizontal (theory) Vertical (theory)
-45 0
20
40
60
Time (ms)
80
100
120 30
Imperfections, USPAS, January 2008
Two vs. four families for chromaticity correction
Two families of sextupoles not enough for correcting off-momentum optics functions’ distortion and second order chromaticity Solutions: Place sextupoles accordingly to eliminate second order effects (difficult) Use more families (4 in the case of of the SNS ring) Large optics function distortion for momentum spreads of ±0.7%,when using only two families of sextupoles 31 Absolute correction of optics beating with four families
References
Imperfections, USPAS, January 2008
O. Bruning, Accelerators, Summer student courses, 2005. H. Wiedemann, Particle Accelerator Physics I, Springer, 1999. K.Wille, The physics of Particle Accelerators, Oxford University Press, 2000.
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